What are the rules of transformation - specifically, of dilation, rotation, reflection and translation?

Answer 1

The rules for translation (shift), rotation, reflection and dilation (scaling) on a two-dimensional plane are below.

Rules of translation (shift)

You need to choose two parameters: (a) direction of the translation (straight line with a chosen direction) and (b) length of the shift (scalar). These two parameters can be combined in one concept of a vector.
Once chosen, to construct an image of any point on a plane as a result of this transformation, we have to draw a line from this point parallel to a vector of translation and, in the same direction as chosen on the vector, move a point along this line by a chosen length.
1. Rules of rotation
  
  You need to choose two parameters: (a) center of rotation - a fixed point on a plane and (b) angle of rotation.
  Once chosen, to construct an image of any point on a plane as a result of this transformation, we have to connect a center of rotation by a vector with our point and then rotate this vector around a center of rotation by an angle congruent to a chosen angle of rotation.
  1. Rules of reflection
    
    You need to choose only one parameter - the axis (or line) of reflection.
    Once chosen, to construct an image of any point on a plane as a result of this transformation, we have to drop a perpendicular from our point onto an axis of reflection and extend it to the other side of the plane beyond this axis by the same distance.
    1. Rules of dilation ( scaling)
      
      You need to choose two parameters - (a) center of scaling and (b) factor of scaling.
      Once chosen, to construct an image of any point on a plane as a result of this transformation, we have to connect a center of scaling with our point and stretch or shrink this segment by a scaling factor, leaving the center of scaling in place. Factors greater than 1 will stretch the segment, factors from 0 to 1 are shrinking this segment. Negative factors reverse the direction of a segment to opposite side from the center.

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Answer 2

Sadie Allen

The rules for transformations are as follows:

Dilation:
- Each point in the original figure is stretched or compressed away from or toward a fixed point called the center of dilation.
- The scale factor determines the amount of stretching or compression.
- If the scale factor is greater than 1, the figure is enlarged.
- If the scale factor is between 0 and 1, the figure is reduced.
- If the scale factor is negative, the figure is reflected across the center of dilation.
Rotation:
- Each point in the original figure is rotated about a fixed point called the center of rotation.
- The angle of rotation determines the amount of rotation.
- Positive angles rotate counterclockwise, while negative angles rotate clockwise.
Reflection:
- Each point in the original figure is reflected across a fixed line called the line of reflection.
- The line of reflection acts as a mirror, with points on one side being reflected onto the other side.
Translation:
- Each point in the original figure is moved a certain distance horizontally and/or vertically.
- The distance and direction of the movement are determined by a vector, which specifies the horizontal and vertical shifts.

These rules are fundamental in understanding how shapes can be transformed in geometry.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.